Structural Instability and Magnetism of Superconducting KCr$_3$As$_3$

TL;DR

This study reveals a lattice instability in KCr₃As₃ that affects its electronic and magnetic properties, providing insights into its potential unconventional superconductivity.

Contribution

It uncovers a lattice distortion in KCr₃As₃ that influences its electronic structure and magnetic behavior, highlighting the role of structural instability in superconductivity.

Findings

Lattice instability causes distortion of Cr wires.

Distortion enhances one-dimensional electronic features.

Distortion suppresses magnetic tendencies.

Abstract

We find a lattice instability in the superconductor KCr$_3$As$_3$, corresponding to a distortion of the Cr metallic wires in the crystal structure. This distortion couples strongly to both the electronic and magnetic properties, in particular by making the electronic structure much more nearly one-dimensional, and by shifting the compound away from magnetism. We discuss the implications of these results in the context of the possibly unconventional superconductivity of this phase.

Full text

I Introduction

The superconductivity of the Cr-based materials, exemplified by K2Cr3As3 and KCr3As3, has attracted recent interest. Mu et al. (2018a, 2017); Bao et al. (2015a); Tang et al. (2015a); Wang et al. (2015); Yang et al. (2015) This arises in part from the possible connections with the Fe-based superconductors, and from unique features of these materials. These include, non-centrosymmetry, proximity to magnetism and quasi-one-dimensional structural features in the crystal structure. A variety of related compounds have been found, including both Cr and Mo compounds. The structures of these materials, $A_{x}M_{3}$As3 (with $A$ = Na, K, Rb, Cs, $M$=Cr, Mo) contain metal-arsenic tubes separated by alkali-metal cations. Mu et al. (2018a); Pang et al. (2016); Zhou et al. (2017); Adroja et al. (2015); Kong et al. (2015); Zhi et al. (2015); Adroja et al. (2017); Tang et al. (2015b); Mu et al. (2018b); Zhao et al. (2018) The superconducting transition temperature of K2Cr3As3 is $T_{c}$=6.1 K,Bao et al. (2015a) and decreases with pressure. This has been associated with changes in metal-pnictogen bond angles analogous to the Fe-based pnictide superconductors.Wang et al. (2016) These compounds occur in hexagonal crystal structures, as mentioned, with Cr (or Mo) wires running along the $c$-axis direction. These wires consist of perfect Cr3 triangles stacked along $c$, and coordinated on the outside by As.

Theoretical studies of K2Cr3As3 show a complex Fermi surface structure with three-dimensional and one-dimensional sheets. Wu et al. (2015); Subedi (2015) The one-dimensional sheets have been seen in angle resolved photoemission (ARPES) experiments. Watson et al. (2017) There is also a suppression of spectral weight near the Fermi energy following Tomonaga-Luttinger liquid behavior, characteristic of a one-dimensional electronic system. The electronic structure near the Fermi energy is mainly from the Cr-3d electrons.Jiang et al. (2015); Wu et al. (2015); Subedi (2015) Theoretical calculations find weak magnetic instabilities at the zone center and competing magnetic states. Wu et al. (2015) This would be consistent with triplet superconducting state, following similar arguments to Sr2RuO4. Rice and Sigrist (1995); Machida et al. (1996); Mazin and Singh (1997) This magnetic tendency is a consequence of the high electronic density of states at the Fermi level, $N(E_{F})$, associated with the multiple Fermi surface sheets.Jiang et al. (2015); Wu et al. (2015) A triplet state was additionally suggested based on a Hubbard model for the Cr wires. Zhong et al. (2015) Nearness to ferromagnetism has been supported by NMR measurements in Rb2Cr3As3 and K2Cr3As3, which find an increasing spin-lattice relaxation rate with cooling. The Hebel-Slichter peak is absent, suggesting the possibility of an unconventional superconducting state. Zhi et al. (2015); Yang et al. (2015) However, lack of a Hebel-Slichter peak may also be a consequence of strong coupling in a conventional scenario, such as that proposed by Subedi. Subedi (2015) In any case, spin fluctuations are also seen in neutron scattering data. However, they are much weaker than in, e.g. Fe-based superconductors, and are not at the zone center. Taddei et al. (2017) Muon spin relaxation ($\mu$SR) experiments are reported to be consistent with an unconventional singlet state. Adroja et al. (2015, 2017) However, it was noted that this may also be consistent with a fully gapped conventional $s$ wave state. Adroja et al. (2015) In contrast, penetration depth measurements suggest nodes in the gap. Pang et al. (2015) However, the situation is complicated by the non-centrosymmetric crystal structure of K2Cr3As3, which can mix singlet and triplet states.

Here we focus on the KCr3As3 compound, which has a somewhat lower electron count in the Cr3As3 tubes, but is otherwise reported to be very similar in superconducting properties to compounds in the K2Cr3As3 family. Bao et al. (2015b); Zhang et al. (2019); Mu et al. (2017) Similar to K2Cr3As3, the ground state reported by theoretical calculations is magnetic, and the Fermi surface shows one-dimensional and three-dimensional sheets. Cao et al. (2015) However, magnetic order is not seen in experiments. Interestingly though, Bao and co-workers reported a possible spin-glass ground state in non-superconducting samples. This was accompanied by a Curie-Weiss behavior of the susceptibility with a fitted effective moment of 0.68 $\mu_{B}$. Bao et al. (2015b) In contrast, experiments on other samples find different behavior, Mu et al. (2017) perhaps related to subtle chemical differences due to different sample preparations. Replacement of K by Rb leads to an increase in critical temperature from $\sim$5 K to 7.3 K, showing that the superconductivity in this family is as robust as that in the K2Cr3As3 compounds. Liu et al. (2017) The compound is reported to occur in a hexagonal TlFe3Te3-type crystal structure, spacegroup $P6_{3}/m$ (number 176), which is centrosymmetric. Bao et al. (2015b); Mu et al. (2017)

Here we show that hexagonal KCr3As3 has a lattice instability. This is related to recent experimental and theoretical findings for hexagonal K2Cr3As3. Taddei et al. (2018) This has strong effects on the magnetic and electronic properties, with implications for the superconductivity.

II Methods

The reported results were obtained within density functional theory (DFT) using the Perdew-Burke-Ernzerhof generalized gradient approximation (PBE GGA), Perdew et al. (1996) and the experimentally reported lattice parameters, $a$=9.0909 Å, $c$=4.1806 Å. The internal atomic coordinates are relaxed by total energy minimization, subject to symmetry. The shortest Cr-Cr bond length in a triangular layer is 2.58 Å, which is almost the same as the distance of 2.57 Å, between closest Cr in adjacent layers with the hexagonal structure.

Phonon calculations were done using the finite difference supercell method using the PHONOPY code. Togo and Tanaka (2015) We used 2$\times$2$\times$4 supercells with underlying DFT calculations performed using the projector-augmented wave method as implemented in the VASP code. Kresse and Joubert (1999); Kresse and Furthmüller (1996) The energy cutoff was 500 eV and a Brillouin zone integration grid spacing of 2$\pi\times$0.032 Å*-1* was used. A phonon instability was found at the $\Gamma$ point. We relaxed the crystal structure allowing the condensation of the unstable modes, first of all using VASP, and then performed a final relaxation using the general potential linearized augmented planewave (LAPW) method, Singh and Nordstrom (2006) as implemented in WIEN2k. Blaha et al. (2001) The LAPW method was used for all the electronic structure and magnetic calculations and for the energies reported here. These calculations were done using well converged basis sets obtained with an interstitial planewave sector cutoff set by $RK_{max}$=9, where $R$ is the smallest LAPW sphere radius and $K_{max}$ is the cutoff. Local orbitals were used for the semicore states. Structure relaxations and energies were obtained including relativity at the scalar relativistic level. Spin orbit was included in the reported electronic structures, and dense converged samplings of the Brillouin zone were employed.

III Results and Discussion

Our calculated phonon dispersions for the experimentally observed hexagonal structure are shown in Fig. 1. We find very strong phonon instabilities in the $k_{z}$=0 plane with four unstable branches, and weaker instabilities at $k_{z}$=1/2. These branches show little dispersion at constant $k_{z}$. This is qualitatively similar to K2Cr3As3, Taddei et al. (2018) but the instabilities in the present case are stronger and there are more unstable branches. This and the weak dispersion of the modes in the basal ($k_{z}$=0) plane imply a strong structural instability of the individual Cr3As3 tubes are consistent with little correlation between tubes.

We explored this structural instability by fully relaxing the atomic coordinates without symmetry, starting from a displacement corresponding to the most unstable phonon at $\Gamma$. This relaxation was done within a single unit cell to explore the effect of the distortion of the Cr3As3 tubes. The actual structure likely has disorder between different tubes similar to K2Cr3As3. Taddei et al. (2018) This is consistent with low dispersion of the unstable modes in the $k_{z}$=0 plane. In particular, flat dispersions mean that low energy differences between different in plane orderings (i.e. between different tubes) are to be expected with a resulting low coherence length for this type of distortion, even though the distortion energy is large with respect to the ideal hexagonal structure.

The relaxation yielded to a structure with spacegroup $P$21/$m$. The distortion consists primarily of a distortion of the Cr3 wires in the crystal structure and is depicted in Fig. 2. Compared with the reported structure with spacegroup $P$63/$m$, it shows three different types of Cr atom and two different Cr-Cr bonds bond lengths, 2.74 Å and 2.42 Å  respectively. The energy gain from the distortion is 0.15 eV per formula unit.

This is a very sizable energy that is well outside the range expected for DFT errors. It also is much larger than the magnetic energies for the ideal structure (see below). This means that the ideal structure cannot be stabilized by magnetism. We also performed phonon calculations for the distorted structure. The resulting phonon dispersion is shown in Fig. 3. As seen, the distorted structure does not exhibit unstable modes. Significantly, centrosymmetry is retained in the distorted structure.

The electronic density of states (DOS) are shown for the two structures in lower panels of Fig. 2. The DOS is mainly derived from Cr $d$ states in the region near the Fermi energy, $E_{F}$. The details of the DOS are changed by the distortion, but the overall shape of the DOS is similar for the distorted and undistorted structures. However, as shown in Fig. 4, there is a clear depletion in the DOS near $E_{F}$ in the distorted structure associated in the removal of a peak at $E_{F}$ for the undistorted structure. At $E_{F}$, the DOS, $N(E_{F})$=9.9 eV*-1* per formula unit for the ideal hexagonal structure, but falls to $N(E_{F})$=2.6 eV*-1* for the distorted monoclinic structure. The Cr $d$ contribution as determined by projection onto the LAPW sphere, radius 2.25 bohr, is 6.5 eV*-1* for the hexagonal structure, which amounts to $\sim$2.2 eV*-1* per atom (both spins). This is slightly above the expected critical value for meeting the Stoner criterion for itinerant magnetism. Janak (1977) The distorted structure with its much lower Cr $d$ contribution of $\sim$0.6 eV*-1* per Cr is far from this condition. Thus the distortion may be expected to strongly affect the magnetic properties.

We did magnetic calculations considering the collinear magnetic structures described by Cao and co-workers. Cao et al. (2015) including the doubling of the unit cell along the $c$-axis, for both the distorted and undistorted structures. For the undistorted structure, our results are qualitatively similar to theirs (note that Cao and co-workers used slightly different hexagonal space group for many of the calculations that they report). We find a magnetic ground state with both the PBE GGA and the local spin density approximation. The lowest energy state in our calculation using the experimental space group ($P6_{3}/m$), Bao et al. (2015b); Mu et al. (2017) consists of Cr layers with alternating spin along the $c$-axis. This is a $\Gamma$-point instability, since the unit cell contains two layers.

Magnetic materials are often classified according to the extent to which they are itinerant or local moment in nature. Moriya and Takahashi (1984); Svanidze et al. (2015) In the local moment limit, stable moments are present on the ions due to intra-atomic interactions. These are subject to inter-site interactions that provide the magnetism through ordering of the moments. These can often be described by the Heisenberg or related spin models. Related to KCr3As3. BaMn2As2 is a material close to this limit. Johnston et al. (2011) The opposite, itinerant limit, is exemplified by materials such as ZrZn2, Y2Ni7 and to a lesser extent elemental Ni. Moriya and Takahashi (1984) In this limit, there are not stable moments, but instead ordering arises from conduction electrons, and in fact magnetic states do not exist for some ordering patterns. This is seen for example from merging of spin-wave excitations seen by neutron scattering into a Stoner continuum at high wavevectors. Mook and Paul (1985) Both of these types of magnetism can be described by DFT calculations, as examplified by work on BaMn2As2, Y2Ni7 and other materials. An et al. (2009); Honer et al. (2017); Singh (2015)

We find that the magnetic instabilities are itinerant in nature since the magnetic moments depend strongly on the particular order and the energy differences between different magnetic orders are on the same scale as that of ordered states relative to the non-spin-polarized energy. For example, the ferromagnetic solution has moments that are less than half of the lowest energy antiferromagnetic solution, i.e. 0.21 $\mu_{B}$/Cr vs. 0.53 $\mu_{B}$/Cr, as measured by the spin moment in an LAPW sphere, radius 2.25 bohr. The energy, with respect to the non-spin-polarized solution, of the ferromagnetic state is -8 meV, while the antiferromagnetic is -19 meV both on a per formula unit basis. Thus in its observed hexagonal structure, KCr3As3 is predicted to be an itinerant antiferromagnet. The non-magnetic ground state observed would then require strong spin fluctuations, as may occur near a quantum critical point in analogy with unconventional superconductors such as the Fe-based materials. Mazin et al. (2008); Bondino et al. (2008) However, with the structure distortion the situation is completely different. In particular, we find no magnetic instabilities in the distorted structure.

The Fermi surfaces for the hexagonal and distorted structures are also very different as shown in Fig. 5. In particular, the prominent three-dimensional Fermi surface in the undistorted structure is completely removed by the distortion. Left are four very one-dimensional Fermi surface sheets. All of these contain holes at the zone center (electrons at $k_{z}$=0.5). The electron-like volumes enclosed are 0.709, 0.510, 0.426 and 0.355 in terms of the Brillouin zone volume. The corresponding nesting vectors of these 1D sheets are along $k_{z}$ and are 0.291, 0.490, 0.426 and 0.355 in units of $2\pi/c$. The band structures are shown in Fig. 6. As seen, the change in Fermi surface is related to an upward shift of the bands giving rise to the 3D sections.

IV Discussion and Conclusions

To summarize, we find strong structural instabilities of the reported hexagonal structure of KCr3As3. These lead to a much more one-dimensional Fermi surface structure, a depletion of $N(E_{F})$ and a suppression of magnetism.

The structure calculated here is an ordered structure with distorted Cr3As3 tubes. Considering the experimental fact that the crystal structure of KCr3As3 is reported to be hexagonal, and the phonon dispersions that suggest low coherence of the distortion between tubes, the likely state at low temperature consists of distorted tubes with inter-tube disorder. This is similar to what is found in K2Cr3As3. The present results show a strong coupling between the distortions and the three-dimensional Fermi surface, while the one-dimensional sheets are left. It would therefore seem likely that disordered distortions would also destroy the three-dimensional Fermi surface. It would be important to verify this experimentally, by direct measurements, e.g. with ARPES, and also by transport measurements, where for example the conductivity anisotropy could be used to probe whether or not the three-dimensional Fermi surface remains. It will also be of importance to perform detailed experimental structural studies, such as those reported for K2Cr3As3 to check whether the actual ground state structure is distorted and to determine the details.

Finally, we note that the results have implications for the superconductivity. While there are no magnetic instabilities found for the distorted structure, it might be supposed that the nesting of one-dimensional Fermi surface sheets would provide spin fluctuations near the nesting vectors. However, these would not provide a mechanism for superconductivity. The reason is that according to the spin fluctuation theory of Berk and Schrieffer, Berk and Schrieffer (1966) spin fluctuations are repulsive for a singlet state and attractive for a triplet. For 1D Fermi surfaces at $k_{z}$ and $-k_{z}$ a repulsive (singlet) interaction would favor opposite sign order parameter on the two sheets, which is not a singlet state, while an attractive interaction (triplet) would conversely favor same sign (singlet) order parameter, which is not a triplet.

Thus the Pauli exclusion principle prevents effective pairing on such 1D sheets based on nesting related spin fluctuations. On the other hand, electron phonon interactions are attractive for both singlet and triplet pairing. Nesting of quasi-one-dimensional sheets is expected to lead to strong electron phonon coupling, which could support a conventional singlet $s$ wave state. In this regard, it will be of interest to search for Kohn anomalies in the phonon dispersions using neutron scattering measurements.

Acknowledgements.

We thank Keith Taddei for helpful discussions. This work was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Award Number DE-SC0019114. Support for work at Shanghai University was provided by the National Natural Science Foundation of China (Grants 51672171 and 51861145315), the National Key Basic Research Program of China (Grant 2015CB921600), the fund of the State Key Laboratory of Solidification Processing in NWPU (SKLSP201703) and the Fok Ying Tung Education Foundation. GX is grateful for support from the China Scholarship Council.